March 14, ( ) March 14, / 52

March 14, 2008 ( ) March 14, 2008 1 / 52

( ) March 14, 2008 2 / 52

1 2 3 4 5 ( ) March 14, 2008 3 / 52

I 1 m, n, F m n a ij, i = 1,, m; j = 1,, n m n F m n A = a 11 a 12 a 1n a 21 a 22 a 2n a m1 a m2 a mn A = (a ij ) m n a ij A. ( ) March 14, 2008 4 / 52

II A A A m n O m n. 2. ( ) March 14, 2008 5 / 52

1 2 3 4 5 ( ) March 14, 2008 6 / 52

1 m n A = (a ij ) m n B = (b ij ) m n, A + B m n (i, j) a ij + b ij ( ) March 14, 2008 7 / 52

(1) A + B = B + A A, B F m n. (2) (A + B) + C = A + (B + C) A, B, C F m n. (3) m n A A + O = O + A = A. (4) A = (a ij ) F m n A = ( a ij ) m n A + ( A) = ( A) + A = O A A A B = A + ( B). ( ) March 14, 2008 8 / 52

2 m, n F m n A = (a ij ) m n F λ m n λa Aλ (i, j) λa ij a ij λ λa = λ(a ij ) m n = (λa ij ) m n Aλ = (a ij ) m n λ = (a ij λ) m n ( ) March 14, 2008 9 / 52

(1) λ(a + B) = λa + λb; (2) (λ + σ)a = λa + σa; (3) (λσ)a = λ(σa); (4) 1 A = A; (5) 0 A = O. ( ) March 14, 2008 10 / 52

I 3 A = (a ij ) m n B = (b ij ) n p AB m p (i, j) AB = (c ij ) m p c ij = n a ik b kj = a i1 b 1j + a i2 b 2j + + a in b nj k=1 ( ) March 14, 2008 11 / 52

II 1 (1) A B (2) A B AB (i, j) A i B j ( ) March 14, 2008 12 / 52

1-2 1. A = (3 0 1), B = 1 2 3 AB BA 2 AB BA ( ) ( ) 2 2 1 1 2. A = B = AB BA 1 1 1 1 3 (1) AB = O A = O B = O (2) AB = AC A B = C ( ) March 14, 2008 13 / 52

(1) A(BC) = (AB)C (2) A(B + C) = AB + AC (A + B)C = AC + BC (3) λ(ab) = (λa)b = A(λB) λ ( ) March 14, 2008 14 / 52

(1) a ij (i j) 0 A = a 11 a mm (2) k A = diag(k, k,, k) B A n AB = BA = kb (3) E = diag(1, 1,, 1) ( ) March 14, 2008 15 / 52

A n A 1 = A; A k+1 = A k A A 0 = E, A 0 ( ) March 14, 2008 16 / 52

(1) A r A s = A r+s (2) (A r ) s = A rs 4 (AB) r A r B r AB = BA (AB) r = A r B r 1 1 0 3. A = 0 1 1 A 100 0 0 1 ( ) March 14, 2008 17 / 52

4 A = a 11 a 12 a 1n a 21 a 22 a 2n a m1 a m2 a mn a 11 a 21 a m1 a 12 a 22 a m2 a 1n a 2n a mn A AT ( ) March 14, 2008 18 / 52

(1) (A T ) T = A (2) (A + B) T = A T + B T (3) (ca) T = ca T (4) (AB) T = B T A T ( ) March 14, 2008 19 / 52

A T = A A A T = A A 5 A B A + B AB 4. A A + A T A A T 5. AA T A T A 6. A AA T = 0 A = 0 ( ) March 14, 2008 20 / 52

1 2 3 4 5 ( ) March 14, 2008 21 / 52

1 n A n B AB = BA = E n, A B A 1 1 A B ( ) March 14, 2008 22 / 52

1 A A 1 (A 1 ) 1 = A 2 A, B n AB (AB) 1 = B 1 A 1 A 1, A 2,, A k n A 1 A 2 A k (A 1 A 2 A k ) 1 = A 1 k A 1 2 A 1 1 ; 3 A c ca (ca) 1 = c 1 A 1 = 1 c A 1 4 A A A T (A T ) 1 = (A 1 ) T ( ) March 14, 2008 23 / 52

2 A ij A = a 11 a 12 a 1n a 21 a 22 a 2n a n1 a n2 a nn A = A A 11 A 12 A 1n A 21 A 22 A 2n A n1 A n2 A nn a ij ( ) March 14, 2008 24 / 52

1 A A A 1 = 1 A A ( ) March 14, 2008 25 / 52

1 1 A, B AB = E A B 2 A AB = AC A B = C 3 Cramer Ax = B A 0 A x = A 1 (Ax) = A 1 B ( ) March 14, 2008 26 / 52

1 A A 2 A = A n 2 A A = A n 1 A n 3 n A A 2 + A E = 0 A + E 4 n A, B A + B = AB A E AB = BA 5 α n 1 α T α = 1 E n Q = E = 2αα T Q ( ) March 14, 2008 27 / 52

1 2 3 4 5 ( ) March 14, 2008 28 / 52

1 5x 2 + 10x 3 = 13 x 1 3x 2 + 7x 3 = 2 x 1 + 2x 2 + 9x 3 = 1 Gauss (1) Ā (2) Ā (3) Ā ( ) March 14, 2008 29 / 52

1. 2. 3. ( ) March 14, 2008 30 / 52

A B B A A B (1) A A (2) A B B A (3) A B B C A C ( ) March 14, 2008 31 / 52

m n A m n 0 B B B B B B B B B B B @ 1 0 0 0 0 1 0 0............ 0 0 1 0 0 0 0 0............ 0 0 0 0 1 C C C C C C C C C C C A m n 1 0 0 m n ( ) March 14, 2008 32 / 52

E n ( ) March 14, 2008 33 / 52

P (i, j) = 1... 1 0 1 row i 1.... 1 1 0 row j 1... 1 ( ) March 14, 2008 34 / 52

1 ( ) P i(c) =... c row i 1... 1 ( ) March 14, 2008 35 / 52

P ( i, j(c) ) = 1... 1 c row i... 1 row j... 1 column i column j ( ) March 14, 2008 36 / 52

P (i, j) 1 = P (i, j) P (i(c)) 1 = P (i(c 1 )) P (i, j(c)) 1 = P (i, j(c 1 )). ( ) March 14, 2008 37 / 52

m n A A m m A A n n ( ) March 14, 2008 38 / 52

m n A m P 1, p 2,, P s A n Q 1, Q 2,, Q t A A m n A, B P 1,, P l, Q 1,, Q t A = P 1 P l BQ 1 Q t m n A, B m P n Q A = P BQ ( ) March 14, 2008 39 / 52

n A A = P 1P 2 P m A n 2n (A, E n ) A n E n E n A 1 ( ) March 14, 2008 40 / 52

1 2 1 A = 2 A = 1 3 1 0 1 1 0 2 4 1 2 1 3 1 0 1 0 2 (A 1 ) ( ) March 14, 2008 41 / 52

1 2 3 4 5 ( ) March 14, 2008 42 / 52

A = (a ij ) m n A kl A kl A A kl A = (A kl ) r s ( ) March 14, 2008 43 / 52

(1) (2) ( ) March 14, 2008 44 / 52

1. 2. A B ( ) March 14, 2008 45 / 52

( ) 1 ( A C A 1 A = 1 CB 1 0 B 0 B 1 ( ) 1 ( A 0 A = 1 0 C B B 1 CA 1 B 1 1 ), ), ( ) March 14, 2008 46 / 52

m n A A 11 A 12 A 1q A 21 A 22 A 2q A =... A p1 A p2 A pq A n m A T 11 A T 21 A T p1 A T A T 12 A T 22 A T p2 =... A T 1q A T 2q A T pq ( ) March 14, 2008 47 / 52

( ) March 14, 2008 48 / 52

( 0 En E m 0 ) ( ) P 0,, 0 E n ( ) ( Em P Em 0, 0 E n P ( Em 0 0 P E n ) ), ( ) March 14, 2008 49 / 52

( A B C D ( ) ( ) ( Em 0 A B A = P C D C + P A E n ) B D + P B ). ( ) March 14, 2008 50 / 52

A, B m n A C 0 B = A 0 C B = A B, C A B 0 = 0 A B C = ( 1)mn A B. ( ) March 14, 2008 51 / 52

( A B 2 C D A C ) A, D B { A D CA D = 1 B, A D A BD 1 C, D 3 A, D D A BD 1 C = A D CA 1 B 4 A = a 2 1 a 1 a 2 + 1 a 1 a n + 1 a 2 a 1 + 1 a 2 2 a 2 a n + 1 a n a 1 + 1 a n a 2 + 1 a 2 n ( ) March 14, 2008 52 / 52